On Convergence to Equilibrium for the Boltzmann Hierarchy

Abstract

We study relaxation of correlated solutions of the hard-cutoff Boltzmann hierarchy toward factorized Maxwellian equilibrium. We prescribe the Maxwellian's mass, momentum, and energy in each full particle coordinate, conditional on the others. Centering removes these conserved modes, but collisions still couple each centered level to the next. To control this forcing uniformly in particle number, we combine adapted second-order Sobolev tensor norms with geometric activity weights and prove $\frac12\frac{d}{dt}E_z(H(t))^2 +(λ-\sqrt{2}C_Γz)D_z(H(t))^2\le0,$ where $H$ is the rescaled centered hierarchy, $E_z,D_z$ measure its energy and dissipation, and the coercivity and collision constants $λ,C_Γ$ are independent of particle number. For small activity $z$, dissipation absorbs the coupling and gives contraction and uniqueness in the hierarchy energy. Time-dependent weights yield level-proportional exponential decay. For global solutions, densities and observable generating functions converge to equilibrium, and connected pair correlations decay at twice the one-particle rate. The estimates apply to existing energy solutions. General existence remains an assumption, while statistical mixtures, including continuous laws, of sufficiently regular one-particle solutions uniformly close to the reference Maxwellian, with common conserved moments, provide examples.